Exercise 3.7

Let $R$ be a ring, and $a,b\in R$. Prove that $-(a+b)=(-a)+(-b)$. Deduce that $(-n)a=-(na)$ equals $n(-a)$ for every nonnegative integer $n$.

Answer

Let $R$ be a ring, and $a,b\in R$. From Exercise 3.6, we have $-(a+b)=(-1)(a+b)=(-1)a+(-1)b=-a+(-b)$. Then by induction,

$$-(na)=-(\underset{n\text{times}}{\underbrace{a+a+\cdots +a}})=\underset{n\text{times}}{\underbrace{(-a)+(-a)+\cdots +(-a)}}=n(-a)$$